Principle of Apertureless Scanning Near-Field Optical Microscopy: On the Way to the Optical Metrology of Nanostructures

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1 Journal of the Korean Physical Society, Vol. 47, August 2005, pp. S166 S174 Principle of Apertureless Scanning Near-Field Optical Microscopy: On the Way to the Optical Metrology of Nanostructures Dominique Barchiesi, Anne-Sophie Grimault, Thomas Grosges, Demetrio Macías and Alexandre Vial Laboratoire de Nanotechnologie et d Instrumentation Optique - CNRS FRE 2671 Universit de Technologie de Troyes 12, rue Marie Curie BP-2060 F TROYES Cedex, France (Received 6 September 2004) Apertureless Scanning Near-Field Optical Microscopy (A-SNOM) is a powerful tool for the observation and engineering of nanostructures in near-field optics (NFO). Nevertheless, the interpretation of the experimental recorded signals raises several difficulties due to the strong dependance of the image formation process on the shape and the optical index of the nanostructure, the properties of the probe and also on the illumination and detection conditions of the experimental configuration used. In order to perform an optical measurement, an accurate model and an inversion procedure are necessary. This last approach may offer a wide variety of applications to the SNOM (e.g. the metrology of nanostructures, the optical, thermal properties, or of optical forces...) In this contribution, we focus our attention on an A-SNOM model for metrological applications in near-field optics. PACS numbers: Fc, Dh, Tp, Fx, De Keywords: Near-field, Metrology, Finite element method, Finite difference time domain, Inverse problem, Numerical modeling, Diffraction, Scattering, Classical electromagnetism, Maxwell s equations I. INTRODUCTION The scanning near-field optical microscopy not only has become, after many years of development, an important branch of the optical microscopy, but also it has found diverse applications in several branches of scientific research. Although several different schemes of scanning near-field optical microscopes have been proposed, there is a particular configuration that makes use of scatters, rather than the tapered optical waveguides employed by more conventional systems, to form the image of a sample [1]. This apertureless or scatter-probe microscope appears to be promising to extend the resolution of nearfield optical microscopes. Nevertheless, the weakness of the signals and the complexity of the image formation process raise several difficulties that must be understood in order to employ A-SNOM systems in nanometrology. For a real metrological application of this kind of microscope, several points have to be carefully studied from a theoretical point of view. First, an accurate model has to be developed to take into account important issues of the image formation process. For example, it has been shown recently that considering the vibration of the scatter-probe and the detection mode employed to measure the scattered signal is critical to model A- SNOM [2]. Also, independently from the method em- dominique.barchiesi@utt.fr ployed for the computation of the electromagnetic field or other physical quantities, metrological applications in A-SNOM could be performed through an inversion procedure to deduce values from the data recorded experimentally (size, optical index...). In this paper we show, through some examples, the importance of employing correct models in the description of experiments in A-SNOM (Sec. II. ). Moreover, we solve the inverse problem of retrieving experimental parameters from A-SNOM data, using evolutionary algorithm (Sec. III. ). To our knowledge this is the first time such an inversion procedure is used in NFO. We present our main conclusions in section IV. II. REQUIREMENTS TO DEVELOP ACCURATE MODELS IN NFO An accurate modeling in Near-Field Optical Microscopy requires careful consideration of certain basic steps. In the following subsections we will illustrate some of them. Subsection II. 1. is an introduction of the direct space models [3] we used in this study: the Finite Element Method (FEM) and the Finite Difference Time Domain (FDTD). Subsection II. 2. is dedicated to some applications in NFO to show why the accuracy of the models has to be carefully controlled. For this purpose, computations of the field enhancement and the -S166-

2 Principle of Apertureless Scanning Near-Field Optical Microscopy Dominique Barchiesi et al. -S167- detection mode of ASNOM, of the heat absorbed by a material and of the optical forces illustrate various faces of the needed accuracy and realism of the models. The subsection II. 2. B. is focussed on the problem of the accuracy in spectroscopic studies with FDTD. In particular, we show that Drude s model may not be sufficient for accurate description of the spectroscopy of nanostructures. 1. Finite Element Method and Finite Difference Time Domain In this subsection, we briefly describe the two methods used in this work to model A-SNOM experiments. The first of them is the Finite Element Method (FEM) which takes advantages of the adaptive non regular mesh for describing the high variations of field. The second one is the Finite Difference Time Domain (FDTD) method which is very useful to model spectroscopy. For the sake of brevity we only show the equations corresponding to a one dimensional geometry. However the generalization to more general geometry can be done straightforward. where ν is a test function defined on L 2 (Ω) (the linear space of the scalar functions ν, being 2-integrable on Ω). This integral formulation enables us to consider the domain of computations as the union of N sub-domains, thus Ω = N i=1 Ω i. The problem is solved on each subdomain and the global solution is then the sum of the solutions in each element. The Eq. (3) is the projection of the solution on a basis of test functions ν. The solution verifies exactly the PDE on each node for the given boundary conditions. The basis of polynomial functions ν gives an approximation of the solution into the element. The numerical implementation of the FEM involves a finite basis of functions and consequently a linear system can be solved by using a classic numerical methods such as Gaussian elimination, Cholesky, LU decompositions or, for large systems through iterative procedures such as bi-conjugate gradient, Lanczos algorithm. This method is particularly adapted for the discretization of complex geometries [2,6]. B. Basics of FDTD A. Basics of FEM Many problems in classical electromagnetic theory can be reduced to the wave equation or Helmholtz s equation if an harmonic time dependance of the form exp(jωt) is assumed in the electromagnetic field [4]. For example, if we consider a geometry defined as a function of the (x,y)-coordinates and, additionally, we assume the fields and the media to be invariant along the z direction, it can be shown that for p polarization the homogenous scalar wave equation in domain Ω has the form: [ ( 1 ɛ ) ] + ω2 c 2 µ H z = 0 in Ω, (1) where ɛ and µ respectively denote the complex relative permittivity and the permeability of the media, ω is the angular frequency, c is the velocity of light in vacuum and H z is the z-component of the harmonic magnetic field which is continuous in Ω (impedance condition [5]). The artificial outer boundary that limits the region of computation is characterized by the first order Sommerfeld s (or running waves) boundary conditions [5]: 1 H z ɛ n = jω c H z on Γ 1, (2) where / n is the outgoing normal derivative operator. The FEM is a direct space method that consists in the resolution of a set of partial differential equations (PDE) with boundary conditions in an open or close domain Ω through the variational, or weak, formulation [5]: Ω [ ( 1 ɛ H z ) + ω2 c 2 µh z ] νdω = 0, (3) The FDTD method is a direct space method originally introduced by Yee in 1966 [7] and now widely used for solving electromagnetic problems in different disciplines. Principles and improvements of this technique have been published in several books [8 10]. It mainly consists in the resolution of the discrete curl-maxwell s equations by iteration over the time. For example, the standard recursion equations for the propagation of the electric and magnetic fields, E and H, with spatial and temporal discretizations x and t along the x direction are given by: H y n+1/2 i+1/2 = H y n 1/2 i+1/2 + t ( Ez n i+1 E z n ) i, (4) µ 0 x E z n+1 i = E z n i + t ( H y n+1/2 ɛ 0 ɛ i x i+1/2 ) H y n+1/2 i 1/2,(5) where ɛ 0 and µ 0 are the permittivity and permeability of free space, ɛ i is the relative permittivity at position x = i x and the value of any field U at position x = i x and for the instant t = n t is given by U n i. The following applications of this model are in 3D case. 2. The NFO problems and the use of accurate models To illustrate the basic requirements for a precise modeling in A-SNOM, we use the two methods described above. The examples will be focussed on the need of controlling the realism or correctness of the models employed

3 -S168- Journal of the Korean Physical Society, Vol. 47, August 2005 (inclusion of the model of detection) and on the accuracy in the computation of the field enhancement to calculate two physical quantities derived from the field: the absorbed heat and the optical forces). Also we study the influence of the model of dispersion in the computation of the spectrum scattered by nanostructures. A. Field enhancement, model of detection, field derived quantities and accuracy control The first issue studied in this part of the work deals with the application of FEM in the computation of the field enhancement in A-SNOM. In Fig. 1 we show an example of the mesh employed to analyze the interaction between the probe and a elliptical metallic nano-particle within a computation domain Ω of size µm 2. The configuration investigated is an A-SNOM, working in reflection mode, illuminated from region z > 0 with a p polarized incident field of wavelength λ 0 = 354 nm at an angle of incidence of θ i = 25. We consider a silicon probe with optical index n = j and radius of curvature of the tip: 36 nm. The object observed is an elliptic Silver particle (n = j) above a glass substrate (n = 1.51). Let us note that the smallest cell size in Fig. 1 is less than 1 nm and it leads to a relative accuracy of 0.01 % on the computation of the field after five successive adaptive refinements. This accurate control of the variation of the shape and the size of each cell in the mesh is critical in situations where high variations of intensity are present (field enhancement for example [11]). In Figs. 1(b-c), the two intensity maps reveal the strong dependance of the intensity on the relative positions of the particle and the probe. In Figs. 1(d-e) the dependance of the electromagnetic field on the media surrounding the particle shows the influence of the substrate and the probe in the optical behavior of the nanostructure. A partial conclusion from the previous results is that the realistic geometry of the probe must be included in the model of A-SNOM if the computation of the intensity map is of interest. The next step to fulfill the requirement for realistic modeling in A-SNOM is to include a model of the lock-in detection of the signal [2]. With reference to the geometry shown in Fig. 1, we present in Fig. 2 the signal as a function of the lateral position of the particle corresponding to the scan process. The numerical simulations make use of the same scanning step as in experiments. In this case the detection is made employing a confocal configuration. The vertical vibration of the probe is used to deduce the lock-in detected signal [6] from the Fourier s transform of the Poynting s vector, integrated over the solid angle that represents the numerical aperture of the objective lens. The dashed and the solid curves in Fig. 2 respectively correspond to the zeroth and first order harmonics of this Fourier s Transform. The zeroth order is the model of the signal detected in the far-field without (d) Fig. 1. Example of adaptive mesh of the probe above an ellipsoid on a glass substrate. Computed intensity is showed for two probe-sample separations (b-c). Intensity maps for the particle alone (d), and for the particle on the substrate (e). The Figs. (b-e) show the influence of all the materials on the intensity map. the lock-in [2], whereas the first order harmonic corresponds to the signal detected through the lock-in at the same frequency of the probe vibration. The difference between these two curves shows that the inclusion of the detection mode is critical. This fact has been verified by comparison with A-SNOM experimental data [2]. An interesting application of the field enhancement occurring at the end of the A-SNOM probe could be the generation of a nanometric phase change in an appropriate medium. This process could be induced by the absorbed electromagnetic power density acting like a heat source Q = 0.5ωɛ 0 I(ɛ) E 2 in a phase-change layer of relative complex permittivity ɛ, illuminated with a laser of frequency ω. The heat source Q could induce a crystallization process in a phase-change material if a given experimentally determined threshold of illumination power (c) (e)

4 Principle of Apertureless Scanning Near-Field Optical Microscopy Dominique Barchiesi et al. -S169- Fig. 2. Detected signal computed using the scan and the metallic probe vibration in the model. The dashed curve corresponds to the average signal along a vertical vibration as a function of the ellipsoid lateral position (scan, see Fig. 1). The solid curve is the signal detected by the lock-in at the frequency of the probe vibration (first harmonic, magnification 30). The position of the probe above the sample is also shown (dotted curve). P is surpassed. To illustrate the previous ideas, let us consider a Ge 2 Sb 2 Te 5 alloy. For this example, the illumination conditions are close to those considered for the geometry shown in Fig. 1. That is, the incident field is a p-polarized laser beam illuminating the sample at an angle of incidence θ i = 65. The power P = 7.3 mw of the incident beam of wavelength λ 0 = 660 nm is included in the model. The illuminated surface is 6 µm 2. The phase-change material is a Ge 2 Sb 2 Te 5 alloy (thickness 30 nm, n = j in crystalline state, and n = j in amorphous state) deposited between a carbondiamon protective layer (thickness 2 nm, n = 1.7) and a ZnS-SiO 2 layer (thickness 30 nm, n = j). This multilayered system is deposited on a glass substrate (n = 1.51). The index of refraction of the silicon probe used to create the nanostructure is n = j and is coated with 25 nm of platinum (n = j). Figure 3 shows the map of Q in the involved media. The size of the phase-change dot (crystalline Ge 2 Sb 2 Te 5 ) is of the same order of magnitude as in preliminary experiments. In this case, the FEM takes advantage of the irregular mesh to describe both the core of the probe (1 µm long) and the protective layer (2 nm thick). Another example of application related to the field enhancement at the end of the probe is the creation of polymer dots by photopolymerization [12]. To understand the polymer formation one may compute the volumetric density of optical force F = ɛ 0 (ɛ 1)E. E in the polymer layer [13,14]. Actually, the polymer dot is produced by photopolymer process but its shape is also related to the forces involved in the experiments. Due to the differential operator, the computation of the optical force needs a more accurate computation of the field value. As the geometry considered in Fig. 1, in this case, cells Fig. 3. Optical heat source. The crystallized dot is not only located below the probe end. The size of the phasechange dot (crystalline Ge 2Sb 2Te 5) is of the same order of magnitude as in preliminary experiments. smaller than 1 nm must be used to compute the field below the probe end with relative accuracy of 0.01 %. From Fig. 4 we deduce that the normal incidence mode induces a force F only 3 times smaller than the reflection mode where the field enhancement is expected to be higher. In the example considered, the probe is dielectric. The normal illumination should induce a hole in the polymer as the force is oriented along the negative z axis. On the contrary, the reflection mode should induce a dot. The comparison of these results with experiments are in process. To close this section, we will show the example of the spectroscopy model in which the FDTD method is suitable. B. Spectroscopy and the influence of the dispersion model The FDTD method has proven to be well adapted for spectroscopic studies [15]. Indeed, accurate results for a full spectrum can be obtained in a single run of the program. Nevertheless, a strong limitation is the requirement of an analytical model of dispersion. Typical laws used for FDTD simulations are Debye s, Lorentz or Drude s dispersion models [15 18]. Also, a modified Debye s law can be used [19]. At least in principle, any dispersion law could be described in terms of a linear combination of Debye s and Lorentz laws [9,20]. An example illustrating that such a combination is necessary in some cases is Drude s model, which only gives a good description of the dispersion function of gold for wavelengths above 650 nm. However, if we intend to model extinction spectra on gold nanostructures for wavelengths between 500 and 1000 nm [21,22], an additional Lorentzian term is needed for an accurate description of the dispersion function. The resulting expression can the be written: ɛ DL (ω) = ɛ ωd 2 ω(ω + iγ D ) ɛ.ω 2 L (ω 2 Ω 2 L ) + iγ Lω,(6) where ω D is the plasma frequency and γ D is the damping coefficient for Drude s model, Ω L and Γ L respectively stand for the oscillator strength and the spectral width

5 -S170- Journal of the Korean Physical Society, Vol. 47, August 2005 Fig. 4. Local optical force in a polymer below a Silicon probe coated with Platinum, illuminated in reflection, normal transmission. The maximum of the force in the polymer is 11.8 pn and 4.1 pn respectively. The gray level shows the optical intensity. of Lorentz oscillators, and ɛ can be interpreted as a weighting factor. Results obtained using Eq. (6) are presented on Fig. 5. In Fig. 5, we show the best fit of the permittivity of gold obtained with Drude s and Drude- Lorentz model, respectively. It can be seen that the real part of the permittivity may be well described by Drude s model, but the description of the imaginary part requires the use of Drude-Lorentz model. Computations of extinction spectra above gold cylinders deposited on a glass substrate are presented in Fig. 5. It can be observed that results obtained using Drude-Lorentz model are more realistic than those obtained using single Drude s model and more consistent with previously published experimental results [21]. The main consequence of using single Drude s model is the presence of strong oscillations in the spectrum for low wavelengths where the imaginary part of the permittivity is not well described (Fig. 5(c)). A slight shift of the resonance may also be observed for different kind of structures. The examples of applications presented in this section show that realistic models of A-SNOM must include the (c) Fig. 5. Real and imaginary parts of permittivity, and, deduced from experimental measurements and calculated with Drude s and Drude-Lorentz models. (c) Spectra computed with the FDTD code, using Drude s (thin curves) and Drude-Lorentz (thick curves) models of dispersion. The calculations performed with Drude-Lorentz model are depicted with a thick line. probe and the experimental parameters. In addition, the complexity of the geometries considered, as well as the quantities derived from the calculation of the field, involve the use of an accurate grid of the computational domain. Once the model is suitable to describe the ex-

6 Principle of Apertureless Scanning Near-Field Optical Microscopy Dominique Barchiesi et al. -S171- accurate description of the experiment considered can be used. In Ref. [25] the inverse problem was reformulated as a non-linear constrained optimization problem through the definition of the objective functional: f(p (c) ) = N i=1 [ s i (z 0 ) s (c) i (z 0 p (c) )] 2, (7) Fig. 6. Algorithm of the inversion procedure. The evolutionary loop shows the main steps of the evolutionary algorithm. The constants µ and λ represent the sizes of the initial and the secondary populations, respectively. periment, we are able to use it to solve the NFO inverse problem, to recover the experimental characteristics of the investigated structure. The following section introduce the resolution of the inverse problem in a simple case. III. INVERSION PROCEDURE FOR METROLOGICAL APPLICATIONS IN NEAR-FIELD OPTICS The relation between an object and its image, obtained with an apertureless near-field optical microscope, is complicated and it cannot be expressed in terms of a transfer function except in particular cases where the interaction between the tip and the sample could be neglected [23, 24]. However, this is no longer the case for an A-SNOM configuration and the need of an alternative inversion scheme to measure characteristics of the sample is evident. Depending on the searched parameters, one of the above described models (the direct models) could be used to solve the inverse problem. In a previous work we have introduced an evolutionary approach for the solution of the inverse problem in near-field optics [25]. In that paper, we use 2D-FDTD to validate the feasibility of the method with theoretical noisy NFO data. The purpose of this section is to confront the inverse method to real experimental data in a very simple case. The main advantage of this numerical technique is its total independence of the method used for the solution of the direct problem. That is, any model providing an where is the sampling step, N is the number of sampling points and p (c) is a vector whose components are the objective parameters to be retrieved. This expression can be interpreted as a measure of the difference between the value of the experimental signal s i at the i- th sampling point and the signal s (c) i predicted through a theoretical model previously established. Thus, the original inverse problem has been reduced to minimize the objective functional f(p (c) ). At least in principle, the minimum sought should be the solution of the problem studied. In this work, we apply an inversion procedure analogous to the one described in Ref. [25] (see Fig. 6) for the identification of experimental parameters from near-field data. Although the principles of the evolutionary algorithm used to optimize Eq. (7) have been fully described elsewhere [26], we summarize the main steps of the method in Fig. 6. The secondary population of parameters P (g) λ is generated at each iteration of the algorithm through the recombination and mutation operators. At this stage of the exposition an example is convenient to illustrate and to assay the performance of the inversion scheme proposed. Because of the iterative nature of the method, we will focus our attention on a very specific example where the modeling of the experimental set up can be done employing a simple analytical representation. Although this choice is made to keep the computation problem in a manageable size, it must be noted that it is not restrictive and a more general model could be used (e.g. one of those described in Sec. II. ). Recently, a study of the interferometric effect present throughout the image formation process in Apertureless Scanning Near-Field Optical Microscopy (ASNOM) has been done employing simple experimental records and a related analytical model [27]. In their analysis, Aubert et al make use of approach curves recorded above a sample (nanostructures of gold deposited on the flat interface between two media, glass and air, illuminated in total internal reflection). A lock-in configuration is used to improve the signal-to-noise ratio. The model proposed in Ref. [27] explains the origin of the pattern of fringes observed in the A-SNOM images as an interference process between the background field E g scattered by the sample and the field E t exp(jφ t ) scattered by the probe end. The detected signal is written as:

7 -S172- Journal of the Korean Physical Society, Vol. 47, August 2005 s(z 0, p) Et 2 exp( 2z 0/D p)i 1(2A/D p) + 2 cos(φ t) E t. E g. exp( z 0/D p)i 1(A/D p) + d (8) E t. E g Et exp( 2z0/Dp)I1(2A/Dp) + 2 cos(φt) exp( z0/dp)i1(a/dp) E g + d, (9) where Φ t is the phase between the two fields E g and E t, A is the amplitude of vibration of the tip, I 1 is the modified Bessel function of the first kind [28] and d is an additive term, not considered in the original model in Ref. [27], whose role will be discussed later. The decay length D p of a Fresnel s wave above a prism of refractive index n is a function of the angle of incidence θ i and of the wavelength λ 0 [4]: D p = λ 0 2π n 2 sin 2 (θ i ) 1 with θ i > arcsin(1/n). (10) For consistency with the notation employed in Eq. (7), we have deliberately expressed the functional dependance of Eqs. 8 and 9 in terms of the vector p = (θ i, Φ t, A, E t, E g, d) to clearly identify the physical parameters of interest to be retrieved. Let us precise that the experimental approach curves are obtained from the ASNOM apparatus as shown in Fig. 7. A prism of index n = 1.53@680 nm, with adapted angle, is used to generate Fresnel s waves above the sample. Thus, the corresponding critical angle is θi c = arcsin(1/n) = If θ i > θi c total internal reflection is obtained. Interference patterns are observed by scanning the probe above the prism. The approach curves are recorded using a lock-in detection that involves the vibration of the probe with an amplitude A around the vertical position z 0. The position of the probe above the sample varies along the z direction within the interval [z 0 A, z 0 + A] for z 0 > A. On the other hand, if z 0 < A (taping mode) the position of the probe varies within [0, 2z 0 ] and the detected signal is expected to increase as a function of z 0. This property of the signal can be observed on the left side of both, the experimental and theoretical curves in Fig. 8. The curves in Fig. 8 are the same as in Fig. 7 in Ref. [27]. The theoretical curve shown in this work was computed through Eq. (8) considering the respective parameters provided in Ref. [27] and the additive value d found with the inversion procedure. The visual inspection of Figs. 8 reveals that the agreement between the experimental approach curve and that generated by the inversion procedure is better than with the curve obtained with the theoretical parameters from Ref. [27], indicated in the first row of Table 1. Let us note that this values were determined by visual comparison between the experimental and theoretical curves, and by a priori knowledge on the experimental conditions. The parameters found with the evolution strategy for the dark fringe are also shown in the second row of Table 1. The recovering of the parameters (θ i, Φ t, A, E t, E g, d) are satisfactory as the relative errors shows a good agreement excepted for the vibration amplitude A that differs strongly from the theoretical one. This behavior is due to the high level of noise in the experimental data. Actually, the change of regime described above is not very apparent in the curves in Fig. 8. Therefore, the precise determination of the amplitude may be difficult. Another explanation may be the roughness of the model used. The probe is not included in the analytical model and unfortunately, a more accurate model cannot be used at the present time, because of the rapid increase of the computation time. Convergence of the method is fast (9 min) and the value of the objective function after 5000 steps is smaller than The initial population size is λ = 100 and Fig. 7. Principle of the experimental setup considered in Ref. [27]. Decay length of the Fresnel s wave above the prism as a function of the angle of incidence. The gray zone shows the sensitivity of D p in the considered experimental setup as a function of the angle of incidence θ i.

8 Principle of Apertureless Scanning Near-Field Optical Microscopy Dominique Barchiesi et al. -S173- Table 1. Parameters used in Ref. [27] and parameters recovered with the evolutionary inversion procedure: p = (θ i, Φ t, A, E t, E g, d). CT is the computation time. Parameter θ i ( ) Φ t ( ) A (nm) E t / E g E t (arb. units) d (arb. units) f(p (c) ) CT Model (Ref. [27]) Recovered min Relative error (%) the secondary one is µ = 14 [25]. Therefore, 100 computations of the direct problem are needed at each step, therefore, a more complete model should be improved to decrease its computing time. The value of the recovered E t / E g ratio is 1.89 to be compared to the theoretical value of 1.8. Let us note that the parameters E t and E g are not measured in the experiment, the values employed for the model in Ref. [27] are deduced from the visual agreement between the theoretical and the experimental curves. The additive term d serves to achieve a better fit of the experimental signal s i in spite of the fact that it is expressed in arbitrary units due to the arbitrary gain adjustment of the lock-in. This term was not included in the discussion of Ref. [27]. Although it is clear that much can be still done to improve the accuracy and the efficiency of the inversion method, the results presented in this section show that the evolutionary strategy employed is well suited for the retrieval of parameters directly from information acquired experimentally. IV. CONCLUSION In this contribution we have described the main requirements to enable metrological applications of NFO microscopes. The first of these conditions is the necessity of accurate and realistic models to describe the experiments. The control of accuracy is facilitated by the use of adaptive and non regular mesh. We have shown that the realistic geometry of the probe, the detection mode and an accurate dispersion law have to be included in the models. We have successfully applied an evolutionary approach to recover experimental parameters from A-SNOM data. In the future, the models that we have introduced in this work will be included within the inversion procedure proposed. Under these conditions, metrological applications of SNOM will be available. In order to reduce the computation time of the inverse problem resolution, parallel algorithms and other inversion algorithms will be developed. Fig. 8. Experimental (solid curve), theoretical (dashed curve) and inversion procedure-generated (thick solid curve) approach curves. Fitness behavior as a function of the number of iterations. The parameters used for the theoretical curve are the same as those in Ref. [27] (Table 1). The probe is assumed to be above a dark fringe. The best parameters found by the evolution strategy lead to a good fit of the experimental curve. ACKNOWLEDGMENTS D. Macías acknowledges Le Ministre de la Jeunesse, de l Education Nationale et de la Recherche as well as the Conseil Rgional de Champagne-Ardennes for financial support. Th. Grosges was supported by grant 35935E from the Conseil Rgional de Champagne- Ardennes. We are grateful to the experimentalist members of our laboratory for providing the reference data.

9 -S174- Journal of the Korean Physical Society, Vol. 47, August 2005 REFERENCES [1] R. Bachelot, P. Gleyzes and A. C. Boccara, Microsc. Microanal. Microstruct. 5, 389 (1994). [2] R. Fikri, Th. Grosges and D. Barchiesi, Opt. Lett. 28, 2147 (2003). [3] Ch. Girard and A. Dereux, Rep. Prog. Phys. 59, 657 (1996). [4] M. Born and E. Wolf, Principle of Optics (Pergamon Press, Oxford, 1993). [5] Jianming Jin, The Finite Element Method in Electromagnetics (John Wiley and Sons, New York, 1993). [6] R. Fikri, Th. Grosges and D. Barchiesi, Opt. Commun. 232, 15 (2004). [7] K. S. Yee, IEEE Trans. on Antennas and Propagation, AP-16, 302 (1966). [8] K. Kunz and R. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993). [9] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000). [10] A. Taflove, Advances in Computational Electrodynamics, the Finite-Difference Time-Domain Method (Artech House, Norwood, 1998). [11] P. Royer, D. Barchiesi, G. Lerondel and R. Bachelot, Phil. Trans. R. Soc. Lond. A 362, 821 (2004). [12] R. Fikri, D. Barchiesi, F. H Dhili, R. Bachelot, A. Vial and P. Royer, Opt. Commun. 221, 13 (2003). [13] M. Nieto-Vesperinas, P. C. Chaumet and A. Rahmani, Phil. Trans. R. Soc. Lond. A 362, 719 (2004). [14] K. Sumaru, T. Fukuda, T. Kimura, H. Matsuda and T. Yamanaka, J. Appl. Phys. 91, 3421 (2002). [15] M. C. Beard and C. A. Schmuttenmaer, J. Chem. Phys. 114, 2903 (2001). [16] F. L. Teixeira, Weng C. Chew, M. Straka, M. L. Oristaglio and T. Wang, IEEE Trans. Geosci. Remote Sensing 36, 1928 (1998). [17] Stephen K. Gray and Teobald Kupka, Phys. Rev. B 68, (2003). [18] M. Futamata, Y. Maruyama and M. Ishikawa, J. Phys. Chem. B 107, 7607 (2003). [19] John T. Krug II, Eric J. Sanchez and X. Sunney Xie, J. Chem. Phys. 116, (2002). [20] C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983). [21] N. Flidj, J. Aubard, G. Lvi, J. R. Krenn, M. Salerno, G. Schider, B. Lamprecht, A. Leitner and F. R. Aussenegg, Phys. Rev. B 65, (2002). [22] J. Grand, S. Kostcheev, J.-L. Bijeon, M. Lamy De La Chapelle, P.-M. Adam, A. Rumyantseva, G. Lrondel and P. Royer, Synthetic Met. 139, 621 (2003). [23] J.-J. Greffet, A. Santenac and R. Carminati, Opt. Commun. 116, 20 (1995). [24] D. Barchiesi, J. Microscopy 194, 299 (1999). [25] D. Macías, A. Vial and D. Barchiesi, J. Opt. Soc. Am. A 21, 1465 (2004). [26] H. P. Schwefel, Evolution and Optimum Seeking (John Wiley & Sons Inc., New York, 1995). [27] S. Aubert, A. Bruyant, S. Blaize, R. Bachelot, G. Lerondel, S. Hudlet and P. Royer, J. Opt. Soc. Am. A 20, 2117 (2003). [28] J. N. Walford, J. A. Porto, R. Carminati, J. J. Greffet, P. M. Adam, S. Hudlet, J. L. Bijeon, A. Stashkevitch and P. Royer, J. Appl. Phys. 89, 5159 (2001).